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Discrete Laplacians and discrete magnetic Schrödinger operators feature in many different areas of mathematics. They are used in combinatorics and computer science, appear as discretizations of (pseudo-)differential operators on Riemannian manifolds, serve as toy models for Hamiltonians in mathematical physics and play an important role in the study of random walks just to name a few. Even though discrete operators are used for very different means, their basic structure is always the same. Given a graph G = (X,E) consisting of vertices X and edges E one defines the discrete Laplacian ∆ acting on functions on X (up to a sign) by

On the Existence and Uniqueness of Self-Adjoint Realizations of Discrete (Magnetic) Schrödinger Operators